Determination of Statistical Error Bounds and Uncertainty Measures for Estimates of Noise Power Spectral Density

ABSTRACT

Systems/methods for computing a power spectral density estimate for a noise signal. Where the noise signal appears in two channels (a single channel), n successive data acquisitions from the two channels (the single channel) are used to compute n respective cross (power) spectral densities, which are then averaged. The averaged cross (power) spectral density may then be smoothed in the spectral domain. The magnitude of the smoothed cross (power) spectral density comprises an estimate for the noise power spectral density. An effective number of independent averages may be computed based on the number n, the time-domain window applied to the acquired sample sets, the amount of overlap between successive sample sets, and the shape of the frequency-domain smoothing function. A statistical error bound (or uncertainty measure) may be determined for the power spectral density estimate based on the effective number of averages and the averaged single-channel and cross-channel spectral estimates.

FIELD OF THE INVENTION

The present invention relates to the field of noise estimation, and moreparticularly, to mechanisms for estimating the power spectral density ofa noise signal and for determining the uncertainty associated with suchestimates.

DESCRIPTION OF THE RELATED ART

Electronic devices generate random noise. It is important to be able toestimate the power spectral density of the random noise and to estimatethe uncertainty associated with such estimates. One prior art method fordetermining the power spectral density estimate is to acquire a seriesof sets of samples of the noise signal, compute respective powerspectral densities, and average the computed power spectral densities.Where the noise appears in each of two signal channels, an alternativemethod for determining the power spectral density estimate includes:acquiring a series of two-channel data sets, where each two-channel dataset includes a set of samples from the first channel and a set ofsamples from the second channel; computing respective cross spectraldensities, averaging the cross spectral densities. The uncertainty ofthe power spectral density estimate depends on the number of data setsincluded in the average. Driving the uncertainty down to an acceptablysmall value may require a large number of data sets.

SUMMARY

In one embodiment, a computational method for estimating a powerspectral density of a target noise signal may involve the followingactions. The method involves vector smoothing, i.e., the operation ofsmoothing a complex-valued mean cross-spectral density in the frequencydomain.

The method may include performing n iterations of a set of operations inorder to obtain n corresponding complex-valued cross spectral densitiesbetween a signal a(t) from a first channel and a signal b(t) from asecond channel. The signal a(t) is a sum of a first interfering noisesignal and the target noise signal. The signal b(t) is a sum of a secondinterfering noise signal and the target noise signal. The integer n isgreater than one. The set of operations includes: acquiring a set ofsamples of the signal a(t); acquiring a set of samples of the secondsignal b(t), where the set of samples of the signal a(t) and the set ofsamples of the signal b(t) are acquired simultaneously; and computing acomplex-valued cross spectral density between the sample set of thesignal a(t) and the sample set of the signal b(t).

The method may also include averaging the n complex-valued crossspectral densities to obtain a complex-valued averaged cross spectraldensity.

The method may also include smoothing the complex-valued averaged crossspectral density in the frequency domain to obtain a complex-valuedsmoothed cross spectral density. The estimate of the power spectraldensity of the target noise signal is based on the complex-valuedsmoothed cross spectral density (e.g., by computing the magnitude of thesmoothed cross spectral density).

The method may also include storing the estimate of the power spectraldensity of the target noise signal in a memory. The power spectraldensity estimate may be displayed as a graph on a display screen.

In another embodiment, a computational method for estimating a powerspectral density of a target noise signal may involve the followingactions.

The method may include smoothing a complex-valued averaged crossspectral density to obtain a complex-valued smoothed cross spectraldensity. The complex-valued averaged cross spectral density is anaverage of n complex-valued cross spectral densities, where n is aninteger greater than one. Each of the n complex-valued cross spectraldensities is computed based on a corresponding one of n two-channelacquisitions. Each of the two-channel acquisitions includes anacquisition of a set of samples of a signal a(t) from a first channeland a simultaneous acquisition of a set of samples of a signal b(t) froma second channel. The signal a(t) is a sum of a first interfering noisesignal and the target noise signal. The signal b(t) is a sum of a secondinterfering noise signal and the target noise signal.

The method may also include determining an estimate of the powerspectral density of the target noise signal based on the complex-valuedsmoothed cross spectral density. The estimate of the power spectraldensity of the target noise signal is stored in a memory.

In one embodiment, a computational method may include determining astatistical error bound at each frequency for an estimate of a powerspectral density of a target noise signal y(t). The power spectraldensity estimate may be determined by averaging n complex-valued crossspectral densities to obtain a complex-valued averaged cross spectraldensity. (In some embodiments, the determination of the power spectraldensity estimate may also include spectrally smoothing thecomplex-valued averaged cross spectral density.) The n complex-valuedcross spectral densities are computed based on n respective two-channeldata sets. Each two-channel data set includes a set of samples of asignal a(t) acquired from a first channel and a corresponding set ofsamples of a signal b(t) acquired from a second channel. Each set ofsamples of the signal a(t) and the corresponding set of samples of thesecond signal b(t) are acquired over the same interval of time. Thesignal a(t) is a sum of a first interfering noise signal and the targetnoise signal y(t); the signal b(t) is a sum of a second interferingnoise signal and the target noise signal y(t).

The action of determining the statistical error bound may includecomputing the statistical error bound based on an expression of the form√{square root over (A²+T²)}, where A depends on the number n, and whereT depends on the number n and a coherence function associated with thecomplex-valued averaged cross spectral density.

In situations where the two-channel data sets are overlapped in time,the action of determining the statistical error bound may includecomputing an effective number of independent averages corresponding tothe power spectral density estimate based on the number n and therelative amount of time overlap between successive ones of thetwo-channel data sets. Thus, the value A may be computed using thiseffective number, and the value T may be computed using this effectivenumber and the coherence function.

In the case where the power spectral density estimate is determinedusing spectral smoothing (in addition to the averaging of the ncomplex-valued cross spectral densities), the action of determining thestatistical error bound may include computing an effective number ofindependent averages corresponding to the power spectral densityestimate based on data including: the number n, information specifying atime-domain window that is applied to the sample sets of the signal a(t)and to the sample sets of the signal b(t), and information specifyingthe filter used to perform the spectral smoothing (and perhaps also, therelative amount of time overlap between successive ones of thetwo-channel data sets). In this case, the value A may be computed basedon this effective number, and the value T may be computed based on thiseffective number and on a coherence function associated with thespectrally smoothed cross spectral density.

The method may also include storing the statistical error bound in amemory. The statistical error bound may be displayed along with a graphof the power spectral density estimate.

In one embodiment, a computational method may include determining ameasure of uncertainty for an estimate of a power spectral density of anoise signal from a single-channel measurement. The power spectraldensity estimate may be determined by computing an average of n powerspectral densities derived from n respective sets of samples of thenoise signal. (In some embodiments, the determination of the powerspectral density estimate may also include spectrally smoothing theaverage of the n power spectral densities.) Each of the n power spectraldensities is computed based on a corresponding one of the sample sets.

The action of determining the uncertainty measure includes computing theuncertainty measure based on the number n.

In situations where the sample sets are overlapped in time, the actionof determining the uncertainty measure may include computing aneffective number of independent averages corresponding to the powerspectral density estimate based on the number n and the relative amountof time overlap between successive ones of the sample sets. Theuncertainty measure may be computed based on this effective number.

In the case where the power spectral density estimate is determinedusing spectral smoothing (in addition to the averaging of the n powerspectral densities), the action of determining the uncertainty measuremay include computing an effective number of independent averagescorresponding to the power spectral density estimate based on dataincluding: the number n; information specifying a time-domain windowthat is applied to each of the n sample sets; and the filter used toperform the spectral smoothing (and perhaps also, a relative amount ofoverlap between successive ones of the n sample sets). The uncertaintymeasure may be computed based on this effective number.

The method may also include storing the uncertainty measure in a memory.The uncertainty measure may be displayed on a display screen along witha graph of the power spectral density estimate.

In one embodiment, a computational method includes acquiring ntwo-channel data sets and storing the n two-channel data sets in amemory, wherein for each two-channel data set said acquiring comprises:acquiring a set of samples of a signal a(t) from a first channel,wherein the signal a(t) is a sum of a first interfering noise signal anda target noise signal y(t); and acquiring a set of samples of a signalb(t) from a second channel, wherein the signal b(t) is a sum of a secondinterfering noise signal and the target noise signal y(t), wherein thesample set of the signal a(t) and the sample set of the signal b(t) areacquired over the same interval of time.

The computational method may also include determining a statisticalupper bound for an estimate of a power spectral density of a targetnoise signal y(t). The estimate is determined by averaging ncomplex-valued cross spectral densities to obtain a complex-valuedaveraged cross spectral density. (In some embodiments, the determinationof the power spectral density estimate may also include spectrallysmoothing the complex-valued averaged cross spectral density.) The ncomplex-valued cross spectral densities are computed based respectivelyon the n two-channel data sets.

The action of determining the statistical upper bound may includecomputing the statistical upper bound based on the number n, thecomplex-valued averaged cross spectral density, a first spectral densityand a second spectral density, wherein the first spectral density is anaveraged power spectral density for the signal a(t), wherein the secondspectral density is an averaged power spectral density for the signalb(t).

In situations where the two-channel data sets are overlapped in time,the action of determining the statistical upper bound may includecomputing an effective number of independent averages corresponding tothe power spectral density estimate based on the number n and therelative amount of time overlap between successive ones of thetwo-channel data sets. The statistical upper bound may be computed basedon this effective number, the complex-valued averaged cross spectraldensity, the first spectral density and the second spectral density.

In the case where the power spectral density estimate is determinedusing spectral smoothing (in addition to the averaging of the ncomplex-valued cross spectral densities), the action of determining thestatistical upper bound may include computing an effective number ofindependent averages corresponding to the power spectral densityestimate based on data including: the number n; information specifying atime-domain window that is applied to the sample sets of the signal a(t)and to the sample sets of the signal b(t); and information specifyingthe filter used to perform said spectrally smoothing. In this case, thestatistical upper bound may be computed based on the effective number ofaverages, the spectrally smoothed version of the complex-valued averagedcross spectral density, a spectrally smoothed version of the firstspectral density, and a spectrally smoothed version of the secondspectral density.

The statistical upper bound may be stored in a memory. A graph of thestatistical upper bound may be displayed along with a graph of the powerspectral density estimate of the target noise signal.

BRIEF DESCRIPTION OF THE DRAWINGS

A better understanding of the present invention can be obtained when thefollowing detailed description of the preferred embodiments isconsidered in conjunction with the following drawings.

FIG. 1A is a graph of the spectral density of the input voltage noise ofan OPA627 op amp.

FIG. 1B is a graph of the power spectral density of the phase noise ofan SMA100A signal generator at 800 MHz.

FIG. 2A shows a time-domain noise signal without windowing; FIG. 2Bshows the same signal with a Hann window applied.

FIG. 3A shows a time-domain noise signal with two Hann-windowedacquisitions, and no overlap between the acquisitions; FIG. 3B shows thesame signal with four windowed acquisitions, and 50% overlap betweensuccessive acquisitions.

FIG. 4 is a graph of the relative effective number (n′/n) of averagesdue to overlapping Hann window acquisitions.

FIG. 5 is a graph of actual 95.45% simulation results vs. the EMuncertainty estimate. (See equation (41) for a definition of the EMuncertainty estimate.) (The dotted line depicts ideal performance.)

FIG. 6A illustrates one embodiment of a method for obtaining an estimateof a power spectral density of a target noise signal.

FIG. 6B illustrates another embodiment of a method for obtaining anestimate of a power spectral density of a target noise signal.

FIG. 7A illustrates one embodiment of a method for determining an errorbound for a noise power spectral density estimate based on a series oftwo-channel acquisitions.

FIG. 7B illustrates one embodiment of a method for determining anuncertainty measure for a noise power spectral density estimate based ona series of one-channel acquisitions.

FIG. 8 illustrates one embodiment of a method for determining astatistical upper bound for a noise power spectral density estimatebased on a series of two-channel acquisitions.

FIG. 9 illustrates one embodiment of a computer system 900 that may beused to perform any the method embodiments described herein.

FIG. 10 illustrates an embodiment 1000 of the computer system 900.

While the invention is susceptible to various modifications andalternative forms, specific embodiments thereof are shown by way ofexample in the drawings and are herein described in detail. It should beunderstood, however, that the drawings and detailed description theretoare not intended to limit the invention to the particular formdisclosed, but on the contrary, the intention is to cover allmodifications, equivalents and alternatives falling within the spiritand scope of the present invention as defined by the appended claims.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Terminology

The following is a glossary of terms used in the present document.

Memory Medium—A memory medium is a medium configured for the storage andretrieval of information. Examples of memory media include: variouskinds of semiconductor memory such as RAM and ROM; various kinds ofmagnetic media such as magnetic disk, tape, strip, film, etc.; variouskinds of optical media such as CD-ROM and DVD-ROM; various media basedon the storage of electrical charge and/or other physical quantities;media fabricated using various lithographic techniques; etc. The term“memory medium” may also include a set of two or more memory media whichreside at different locations, e.g., at different computers that areconnected over a network.

Programmable Hardware Element—a hardware device that includes multipleprogrammable function blocks connected via a programmable interconnect.Examples include FPGAs (Field Programmable Gate Arrays), PLDs(Programmable Logic Devices), FPOAs (Field Programmable Object Arrays),and CPLDs (Complex PLDs). The programmable function blocks may rangefrom fine grained (combinatorial logic or look up tables) to coarsegrained (arithmetic logic units or processor cores). A programmablehardware element may also be referred to as “reconfigurable logic”.

Program—the term “program” is intended to have the full breadth of itsordinary meaning. As used herein, the term “program” includes within itsscope of meaning: 1) a software program which is stored in a memory andis executable by a processor, or, 2) a hardware configuration programuseable for configuring a programmable hardware element. Any of themethod embodiments described herein, or, any combination of the methodembodiments described herein, or, any subset of any of the methodembodiments described herein, or, any combination of such subsets may beimplemented in terms of one or more programs.

Software Program—the term “software program” is intended to have thefull breadth of its ordinary meaning, and includes any type of programinstructions, code, script and/or data, or combinations thereof, thatmay be stored in a memory medium and executed by a processor or computersystem. Exemplary software programs include: programs written intext-based programming languages such as C, C++, Java™, Pascal, Fortran,Perl, etc.; graphical programs (programs written in graphicalprogramming languages); assembly language programs; programs that havebeen compiled to machine language; scripts; and other types ofexecutable software. A software program may comprise two or moresubprograms that interoperate in a specified manner.

Hardware Configuration Program—a program, e.g., a netlist or bit file,that can be used to program or configure a programmable hardwareelement.

Computer System—any of various types of computing or processing systems,including a personal computer (PC), a mainframe computer system, aworkstation, a laptop, a tablet computer, a network appliance, anInternet appliance, a hand-held or mobile device, a personal digitalassistant (PDA), a television system, a grid computing system, or otherdevice or combination of devices. In general, the term “computer system”can be broadly defined to encompass any device (or combination ofdevices) having at least one processor that is configured to executeprogram instructions that are stored on a memory medium.

Computational Device—an electronic system that is programmable toexecute arbitrary computational algorithms. The term “computationaldevice” includes within its scope of meaning any of the following: acomputer system, a set of one or more interconnected computer systems, asystem including one or more programmable hardware elements, a systemincluding one or more programmable processors and one or moreprogrammable hardware elements. Any of the various embodiments describedherein may be implemented using a computational device.

The various embodiments disclosed herein may be realized in any ofvarious forms. For example, any of the embodiments disclosed herein maybe realized as a computer-implemented method, a computer-readable memorymedium, or a computer system. Furthermore, any of the embodimentsdisclosed herein may be realized in terms of one or moreappropriately-configured programmable hardware elements (PHEs). Yetfurthermore, any of the embodiments disclosed herein may be realized interms of one or more digital circuits, e.g., custom-designed digitalcircuits.

A computer-readable memory medium is a memory medium that stores programinstructions and/or data, where the program instructions, if executed bya computer system, cause the computer system to perform a method, e.g.,any of a method embodiments described herein, or, any combination of themethod embodiments described herein, or, any subset of any of the methodembodiments described herein, or, any combination of such subsets.

In some embodiments, a computer system may include a processor (or a setof processors) and a memory medium. The memory medium stores programinstructions. The processor is configured to read and execute theprogram instructions from the memory medium. The program instructionsare executable by the processor to implement a method, e.g., any of thevarious method embodiments described herein (or, any combination of themethod embodiments described herein, or, any subset of any of the methodembodiments described herein, or, any combination of such subsets). Thecomputer system may be realized in any of various forms. For example,the computer system may be a personal computer (in any of its variousrealizations), a workstation, a computer on a card, anapplication-specific computer in a box, a server computer, a clientcomputer, a hand-held device, a mobile device, a tablet computer, awearable computer, a computer integrated in a head-mounted display, etc.

In some embodiments, a set of computers distributed across a computernetwork may be configured to partition the effort of executing acomputational method (e.g., any of the method embodiments disclosedherein).

Spectral Noise Density Measurements

Spectral noise density may be measured by acquiring many blocks of datain the time domain, computing the Fourier transform of each acquisition,and averaging the resulting spectra. The averaging serves two purposes:first, to reduce the uncertainty of the density estimates since themeasured noise data, being random, causes the power spectral densitymeasurements themselves to be random; and second, to removecontributions of other interfering but uncorrelated signals when using atwo-channel cross-correlation scheme. This disclosure will quantify howwell resolved the averaged results are and how much statisticalconfidence one can have in the results.

FIG. 1 illustrates two examples of noise density measurements. The firstplot is the input voltage noise density of an op amp. The vertical unitsare in V/√{square root over (Hz)}. Technically, with power spectraldensity the units should be V²/Hz (power per unit of bandwidth), but theformer is easier to interpret and so the square root is taken. Thesecond plot is the phase noise of a signal generator. With phase noise,the usual vertical units are dBc/Hz, which indicate the sideband noisepower per unit bandwidth with respect to the carrier power.

Note that both plots use a logarithmic frequency scale. This is doneconventionally because the noise density is of interest over a widerange of frequencies. Each plot is the result of thousands of averages.

Fourier Transform

Let a(t) be the time-domain signal present in a channel. Then itsFourier transform is:

A(f)=

{a(t)}=∫_(−∞) ^(∞) a(t)exp(−j2πft)dt.  (1)

This integral depends on knowing the signal value a(t) at all times inthe continuum of −∞ to +∞. Of course, a computer-based signalacquisition from a channel is taken only over a finite interval of time,and at a finite number of discrete times, normally evenly-spaced intime. So, if N samples are taken during each interval of sample periodT, let

a _(i) =a(iT)=a(t)

for i=0, 1, 2 . . . , N−1. Then the Discrete Fourier Transform (DFT) of{a_(i)} is:

$\begin{matrix}\begin{matrix}{A_{k} = {A\left( {kf}_{0} \right)}} \\{= {\mathcal{F}\left\{ a_{i} \right\}}} \\{= {\frac{1}{N}{\sum\limits_{i = 0}^{N - 1}{a_{i}{{\exp \left( {{- j}\; 2\pi \; {{ki}/N}} \right)}.}}}}}\end{matrix} & (2)\end{matrix}$

The DFT may be implemented using an FFT.

The a_(i) are real-valued. Thus, the A_(k) are conjugate symmetric infrequency, i.e. A_(k)=A_(N-k)*. This means that half of the spectrum{A_(k)} is redundant. The half of the spectrum above the Nyquistfrequency is often thrown away. In order to preserve the power that theredundant half would have included, the coefficients {A_(k)} may beredefined according to the expressions:

$\begin{matrix}{\begin{matrix}{A_{k} = {A\left( {kf}_{0} \right)}} \\{= {\sqrt{2}\mathcal{F}\left\{ a_{i} \right\}}} \\{{= {\frac{\sqrt{2}}{N}{\sum\limits_{i = 0}^{N - 1}{a_{i}{\exp \left( {{- j}\; 2\pi \; {{ki}/N}} \right)}}}}},{k > 0}}\end{matrix}{{{and}\mspace{14mu} A_{0}} = {\frac{1}{N}{\sum\limits_{i = 0}^{N - 1}{a_{i}.}}}}} & (3)\end{matrix}$

Thus, if {a_(i)} represents a sine wave with an rms amplitude of unity,the magnitude of A_(k) is equal to unity when k corresponds to thefrequency of the sine wave.

The power spectrum is computed as the square of the magnitude of A_(k),or equivalently, the product of A_(k) and its complex conjugate A_(k)*.This product, A_(k)*A_(k), is always real and non-negative, even ifA_(k) is complex-valued.

Power Spectral Density

The power spectral density is computed from A_(k)*A_(k) by normalizingthe power in each frequency bin by the equivalent noise bandwidth (ENBW)of the FFT. In an unwindowed FFT, the ENBW is the sample rate f_(s)divided by the number of points in the FFT and is equal to the bin widthof the frequency bins in Hz. Hence the power spectral density, denotedĜ_(aa), is computed as:

$\begin{matrix}{{{\hat{G}}_{aa}(k)} = {\left( \frac{N}{f_{s}} \right)A_{k}^{*}{A_{k}.}}} & (4)\end{matrix}$

Often, the acquired time-domain data is windowed to improve spectraldynamic range by multiplying the data by the time-domain sequence{w_(i)} before the FFT is computed. The window may have an effectivegain (coherent gain, or CG) other than unity, and so the data going intoor coming out of the calculation of Ĝ_(aa) may be scaled by CG. Thewindow also normally has an ENBW of greater than one frequency binwidth, and so the PSD may also be scaled accordingly.

The value for CG is calculated as:

$\begin{matrix}{{{CG} = {\frac{1}{N}{\sum\limits_{i = 0}^{N - 1}w_{i}}}},} & (5)\end{matrix}$

and ENBW is calculated as:

$\begin{matrix}{{{ENBW} = {\frac{1}{({CG})^{2}N}{\sum\limits_{i = 0}^{N - 1}w_{i}^{2}}}},} & (6)\end{matrix}$

where ENBW is measured in FFT bin widths. CG normally varies between0.25 and 1.00. (However, windows having values of CG outside that rangemay be used as well.) ENBW normally varies between 1.00 and 3. (However,windows having values of ENBW outside that range may be used as well.)For the commonly-used Hann window (sometimes incorrectly called a“Hanning” window), CG=0.5 and ENBW=1.5. FIG. 2A shows a time-domainnoise signal without windowing. FIG. 2B shows the same signal with aHann window applied.

With these two constants (CG and ENBW), the power spectral density (PSD)for a windowed acquisition may be calculated as:

$\begin{matrix}{{{\hat{G}}_{aa}(k)} = {\frac{1}{{{ENBW}({CG})}^{2}}\left( \frac{N}{f_{s}} \right)A_{k}^{*}{A_{k}.}}} & (7)\end{matrix}$

Cross-Spectral Density

In some circumstances, the noise signal of interest occurs in twochannels. Thus, a signal {a_(i)} may be acquired from one of thechannels, and a signal {b_(i)} may be acquired from the other channel.The signals {a_(i)} and {b_(i)} may be subjected to cross-correlation,with the goal being to extract the spectral data (due to the noisesignal of interest) that is common to the two acquired channels and toeliminate the unrelated (uncorrelated) noise that might be added by theacquisition channels. The cross-spectral density (CSD) of the signals{a_(i)} and {b_(i)} may be computed according to the expression:

$\begin{matrix}{{{\hat{G}}_{ab}(k)} = {\frac{1}{{{ENBW}({CG})}^{2}}\left( \frac{N}{f_{s}} \right)A_{k}^{*}{B_{k}.}}} & (8)\end{matrix}$

Note that the CSD comprises a sequence of complex numbers.

Averaging

The hats in Ĝ_(aa) and Ĝ_(ab) are used to denote that they are only(noisy) estimates of their respective underlying true spectraldensities, G_(aa) and G_(ab). Both power spectral density estimates andcross-spectral density estimates can be improved by averaging. In otherwords, a number of the estimates Ĝ_(aa), corresponding to respectiveacquisitions of the signal a(t), may be averaged to obtain an improvedestimate of the spectral density G_(aa). Similarly, a number of theestimates Ĝ_(ab), corresponding to respective acquisitions of the signalpair {a(t),b(t)}, may be averaged to obtain an improved estimate of thespectral density G_(ab). (An acquisition of the signal pair {a(t),b(t)}comprises an acquisition of the signal a(t) and a parallel acquisitionof the signal b(t).)

In the first case, the estimate Ĝ_(aa) is always real, and the mean ofthe probability density distribution of Ĝ_(aa) is in fact G_(aa). Weneed only average the PSDs of successive acquisitions to improve theestimate of G_(aa) until we have sufficient statistical confidence inthe result.

The case of G_(ab) is not so simple. At least one of the acquisitions{a_(i)} and {b_(i)} is presumed to have information that is not presentin the other. It is also possible that each of the acquisitions {a_(i)}and {b_(i)} has information that is not present in the other. Becausethe values Ĝ_(ab)(k) are vector quantities (i.e., complex numbers),averaging over multiple acquisitions can reduce the contributions ofthese uncorrelated signals until the magnitude of the vector average iswell below the average of the magnitude. Expressed mathematically,

| {circumflex over (G)}_(ab) |≦ |Ĝ _(ab)|,  (9)

where the bar over a quantity denotes the mean over some number ofaverages. (In other words, Ĝ_(ab) represents an average taken over anumber of instances of the spectrum Ĝ_(ab). Similarly, |Ĝ_(ab)|represents an average taken over the same number of instances of themagnitude spectrum |Ĝ_(ab)|.)

To see this, let us decompose A_(k) and B_(k) into variables X_(1k),X_(2k) and Y_(k), where all three are uncorrelated with each other andwhere

A _(k) =X _(1k) +Y _(k) and

B _(k) =X _(2k) +Y _(k).

Signals X_(1k) and X_(2k) represent the noise added respectively by thetwo channels when we are really trying to measure G_(yy), the PSD ofY_(k).

To calculate Ĝ_(ab), we write out the product A_(k)*B_(k) as:

$\begin{matrix}\begin{matrix}{{A_{k}^{*}B_{k}} = {\left( {X_{1k} + Y_{k}} \right)^{*}\left( {X_{2k} + Y_{k}} \right)}} \\{= {{X_{1k}^{*}X_{2k}} + {X_{1k}^{*}Y_{k}} + {Y_{k}^{*}X_{2k}} + {Y_{k}^{*}{Y_{k}.}}}}\end{matrix} & (10)\end{matrix}$

Hence,

Ĝ _(ab) =Ĝ _(x) ₁ _(x) ₂ +Ĝ _(x) ₁ _(y) +Ĝ _(yx) ₂ +Ĝ _(yy),  (11)

and when we average the estimates,

{circumflex over (G)}_(ab) = {circumflex over (G)}_(x) ₁ _(x) ₂ +{circumflex over (G)}_(x) ₁ _(y) + {circumflex over (G)}_(yx) ₂ +{circumflex over (G)}_(yy) .  (12)

Since X_(1k) and X_(2k) are presumed to be uncorrelated with each other,as are X_(1k) and Y_(k) as well as X_(2k) and Y_(k), then as the numberof averages increases (i.e., the number of acquisitions over which themeans are computed), all terms of (12) converge to zero except the last,leaving, in the limit,

{circumflex over (G)}_(ab) = {circumflex over (G)}_(yy) .  (13)

Thus, we can measure G_(yy) by averaging a sufficient number ofacquisitions of Ĝ_(ab). Because of this property, the cross-correlationtechnique is able to resolve spectral information well below thespectral noise floors of the individual measurement channels.

Probability Density Distributions

Let X₁ and X₂ be complex random variables, with

X ₁ =a ₁ +jb ₁ and

X ₂ =a ₂ +jb ₂,

and where a₁, a₂, b₁, b₂ are all standard normal, i.e.,

a ₁ ,a ₂ ,b ₁ ,b ₂ ˜N(0,1).

These kinds of variables are the typical results of Fourier powerspectra when the input is random noise. Let X₁*X₂ denote the average ofn independent samples of X₁*X₂, and let | X₁*X₂ | denote the magnitudeof X₁*X₂ ·| X₁*X₂ | is used to calculate the cross-spectral density ofX₁ and X₂, or the power spectral density of X₁ when X₁=X₂.

Equivalence

Suppose X₁=X₂, as is the case with a PSD computation and with the Ĝ_(yy)term of the G_(ab) cross-spectral density measurement (see equation(12)). Then | X₁*X₂ |=| X₁*X₁ | has a distribution that is proportionalto a chi-square distribution with 2n degrees of freedom, with mean andstandard deviation:

$\begin{matrix}{{{\mu \left\lbrack {\overset{\_}{X_{1}^{*}X_{1}}} \right\rbrack} = 2}{and}} & (16) \\{{\sigma \left\lbrack {\overset{\_}{X_{1}^{*}X_{1}}} \right\rbrack} = {\frac{2}{\sqrt{n}}.}} & (17)\end{matrix}$

As n increases, the distribution converges to a normal distribution(with the same mean and standard deviation).

If a₁, a₂, b₁ and b₂ have a standard deviation other than unity, say s,then both the mean and standard deviation of | X₁*X₁ | scale with s².Hence the ratio of the standard deviation to the mean is always1√{square root over (n)}, and so the amount of variation of the estimateof X₁*X₁ in dB is a function only of n. The decibel equivalent of thestandard deviation of the estimate error becomes:

$\begin{matrix}{{10\; {\log\left( \frac{\sigma + \mu}{\mu} \right)}} = {10\; {\log \left( {{1/\sqrt{n}} + 1} \right)}\mspace{14mu} {{dB}.}}} & (18)\end{matrix}$

Similarly, the equivalent 2σ (95.45%) confidence interval is

$\begin{matrix}{{{{\pm 10}\; {\log\left( \frac{{2\sigma} + \mu}{\mu} \right)}} = {{\pm 10}\; {\log \left( {{2/\sqrt{n}} + 1} \right)}\mspace{14mu} {dB}}},} & (19)\end{matrix}$

but because the actual distribution of | X₁*X₁ | is chi-square and notGaussian, and because of the log transformation, a better estimate ofthe 95.45% confidence interval is

$\begin{matrix}{{\pm 10}\; {\log\left( {\frac{2}{\sqrt{n} - 1} + 1} \right)}\mspace{14mu} {{dB}.}} & (20)\end{matrix}$

This quantity will be used later to estimate the measurement uncertaintydue to taking only a limited number of averages. Note that in order toachieve ±1 dB of uncertainty with 95.45% confidence, we need to take atleast 76 averages. This minimum holds for any spectral powermeasurement.

Independence

Suppose instead that X₁ and X₂ represent independent sources of noise,e.g., as is the case with each of the first three terms on the righthand side of equation (12). Then | X₁*X₂ | has a distribution thatconverges to a Rayleigh distribution as n→∞. The mean and standarddeviation are exactly:

$\begin{matrix}{{{\mu \left\lbrack {\overset{\_}{X_{1}^{*}X_{2}}} \right\rbrack} = {\pi {\prod\limits_{i = 1}^{n}\frac{{2i} - 1}{2i}}}}{and}} & (21) \\{{\sigma \left\lbrack {\overset{\_}{X_{1}^{*}X_{2}}} \right\rbrack} = {\sqrt{\frac{4}{n} - \left( {\pi {\prod\limits_{i = 1}^{n}\frac{{2i} - 1}{2i}}} \right)^{2}}.}} & (22)\end{matrix}$

These two quantities are approximated very well by:

$\begin{matrix}{{{\mu \left\lbrack {\overset{\_}{X_{1}^{*}X_{2}}} \right\rbrack} \approx \sqrt{\frac{\pi}{n + \frac{4}{\pi} - 1}}}{and}} & (23) \\{{\sigma \left\lbrack {\overset{\_}{X_{1}^{*}X_{2}}} \right\rbrack} \approx {\sqrt{\frac{4}{n} - \frac{\pi}{n + \frac{4}{\pi} - 1}}.}} & (24)\end{matrix}$

As stated above, for large n, the distribution converges to a Rayleighdistribution as:

$\begin{matrix}{{{f(y)} = {\frac{ny}{2}^{{- {ny}^{2}}/4}}},{where}} & (25) \\{y = {{\overset{\_}{X_{1}^{*}X_{2}}}.}} & (26)\end{matrix}$

The mean and standard deviation of the limiting Rayleigh distributionare:

$\begin{matrix}{{{\mu \lbrack y\rbrack} = \sqrt{\frac{\pi}{n}}}{and}} & (27) \\{{\sigma \lbrack y\rbrack} = {\sqrt{\frac{4 - \pi}{n}}.}} & (28)\end{matrix}$

Equation (23) and (for large n) equation (27) give us an idea of howmany averages it will take to remove the uncorrelated noise sufficientlyto reveal the correlated noise density. For example, if n is increasedby a factor of 10 (presumably increasing the measurement time by afactor of 10), the mean of the uncorrelated cross-spectral power densitydecreases by a factor of about √{square root over (10)}. Thatcorresponds to a 5 dB improvement in the visible noise floor. So foreach 5 dB improvement in the floor, the measurement will take 10 timesas long; a 10 dB improvement will take 100 times, etc. Ultimately, theavailable measurement time will set a practical limit on how low thenoise floor can be resolved, and the cost of the measurement time willhave to be weighed against the cost of lower-noise measurement channelsin order to achieve a certain level of performance.

Effective Number of Averages

The above-described calculations of the reduction of noise and biaserror as a result of averaging the spectral data assume that the numberof acquisitions averaged, n, actually represents n independentobservations. However, several factors cause the effective number ofindependent observations (referred to herein as “averages”), which wewill call n″, to differ from n. First, the spectra from successiveacquisitions may not be independent, since the acquisitions may havebeen overlapped in time in order not to waste any time-domain data thatwould otherwise be attenuated by the time-domain window near thebeginning and end of an acquisition. Secondly, improvement in the noiselevel and bias error by smoothing the data in the frequency domain(which is weighted averaging across a frequency band) is affected bothby the time-domain window applied to the data before the spectrum iscomputed and by the shape of the frequency-weighting used in thesmoothing procedure.

Overlap

When applying a window to the time-domain data, the data near thebeginning and end of each acquisition tends to be ignored because of theattenuating shape of the window. To make the most use of the finiteamount of time available to acquire data, data from two adjoiningacquisitions may be re-used if there is no time gap between theadjoining acquisitions, forming new, albeit not completely independent,acquisitions. See FIG. 3, which shows (a) a time-domain noise signalwith two Hann-windowed acquisitions, no overlap, and (b) the same signalwith four windowed acquisitions, 50% overlap.

Because some (or most) data appears in more than one acquisition whenoverlapping is used, the acquisitions cannot be considered independent,and the effective number of averages n′ will be less than the actualnumber n. FIG. 4 illustrates this reduction as a function of the amountof overlap when a Hann window is used. As the amount of overlap isincreased, the relative effective number of averages is decreased,although the actual number of averages n increases. For example, at 90%overlap there are 10 times the number of acquisitions than would havebeen taken with no overlap. With a relative effective number of averagesof about 0.2079 at 90% overlap, the effective number of averages n′would be 0.2079n, but n equals 10n_(nol), where n_(nol) is the number ofaverages that could have been taken if no overlap were used, and thus,n′=2.079 n_(nol). For most windows, this gain in effective averagesconverges to a limit as the overlap approaches 100%. This limit is 1/A,where

$\begin{matrix}{{A = {\frac{1}{T}{\int_{2t_{0}}^{2{({t_{0} + T})}}{\left\lbrack {\frac{1}{E\; N\; B\; {W\left( {C\; G} \right)}^{2}T}\left( {w*w} \right)(t)} \right\rbrack^{2}\ {t}}}}},} & (29)\end{matrix}$

and w(t) is the theoretical window shape defined over the time intervaltε[t₀, t₀+T].

For the Hann window,

$\begin{matrix}{{A = {\left\lbrack {\frac{1}{3} + \frac{35}{6\left( {2\pi} \right)^{2}}} \right\rbrack \approx 0.481093}},} & (30)\end{matrix}$

which means the best possible gain in effective averages, from having nooverlap to having complete overlap, is a factor of about 2.079.

For the Hann window, perhaps the most practical overlap is 50%, wherethe effective number of averages n′ would be 0.9475n, which is1.895n_(nol), since twice as many acquisitions were taken. This gain of1.895 is almost as high as the 2.079 limit, but it suffers from only a5.25% decrease in the effective number of averages from the number ofaverages actually taken. As a practical matter, the difference between0.9475 and 1.00 is small enough that the lower figure can be used forboth overlapping and non-overlapping situations, simplifying thenumerical housekeeping in cases where overlapping may not always bepossible due to buffer size limitations.

Spectral Smoothing

The frequency-domain data (e.g., Ĝ_(ab) in the two-channel case, orĜ_(aa) in the one-channel case) may be smoothed. This may be done tocollapse data that's dense along the frequency axis to a more manageableset of frequencies and/or to improve (i.e., increase) the number ofeffective averages. Whenever a time-domain window is used on thetime-domain samples (before the FFT), the noise data in adjacent FFTfrequency bins are not independent. (Multiplication by a window functionin the time domain is equivalent to convolution by the transform of thewindow function in the frequency domain.) So when frequency-domain datais smoothed by averaging, the resulting effective number of averages isnot simply the number of acquisitions used to compute Ĝ_(ab) (or Ĝ_(aa)) times the number of frequency points smoothed—the correlation betweenadjacent frequency bins has to be taken into account. Furthermore, thefrequency-domain envelope of the averaging weights used to perform thesmoothing tends to emphasize data nearest the nominal frequency pointand attenuate data farthest away in frequency. This also causes theeffective number of averages to be less than the simple calculation of aproduct.

To account for the spectral smoothing, let w(t) be the theoretical shapeof the time-domain window, where tε[t₀, t₀+T], t₀ is the beginning timeof the acquisition, and t₀+T is the end. Define

$\begin{matrix}{c_{k} = {{\frac{1}{T}{\int_{t_{0}}^{t_{0} + T}{{w^{2}(t)}\ ^{{- j}\; 2\; \pi \; {{kt}/T}}{t}}}}}^{2}} & (31)\end{matrix}$

as the power of the Fourier series expansion of the square of the windowfunction, with the observation that c₀ is the square of the averagevalue of the square of the window curve.

For the commonly-used Hann window,

c ₀= 9/64=0.140625,

c ₁= 1/16=0.0625,

c ₂= 1/256=0.00390625,

and c _(k)=0, for k>2.

Let H_(i), i=0, 1, 2, . . . , m−1, denote the averaging weights used toperform the spectral smoothing. m is the number of spectral densitypoints to be averaged together. In the two channel case, the mean crossspectral density Ĝ_(ab) is smoothed to obtain a smoothed cross spectraldensity V_(ab). Each point of the smoothed cross spectral density V_(ab)may be computed by applying the averaging weights H_(i) to acorresponding continuous band of uniformly-spaced frequency-adjacentcomplex points in the mean cross spectral density Ĝ_(ab) and thensumming the weighted complex points. In the one channel case, the meanpower spectral density Ĝ_(aa) is smoothed to obtain a smoothed powerspectral density V_(aa). Each point of the smoothed power spectraldensity V_(aa) may be computed by applying the averaging weights H_(i)to a corresponding continuous band of uniformly-spacedfrequency-adjacent real-valued points in the mean power spectral densityĜ_(aa) and then summing the weighted points.

Define

$\begin{matrix}{d_{k} = \left\{ {\begin{matrix}{{\sum\limits_{i = 0}^{m - 1}H_{i}^{2}},} & {k = 0} \\{{2{\sum\limits_{i = 0}^{m - k - 1}{H_{i}H_{i + k}}}},} & {0 < k < m}\end{matrix}{and}} \right.} & (32) \\{a = {\left( {\sum\limits_{i = 0}^{m - 1}H_{i}} \right)^{2}.}} & (33)\end{matrix}$

Then the effective number of independent averages n″ produced by thefrequency-domain (spectral) smoothing of the average of n′ overlappedacquisitions is:

$\begin{matrix}{n^{''} = {\frac{n^{\prime}}{\left\lbrack {\sum\limits_{k = 0}^{m - 1}{\left( {\frac{c_{k}}{c_{0}} + 1} \right)\left( \frac{d_{k}}{a} \right)}} \right\rbrack - 1}.}} & (34)\end{matrix}$

Depending on how many frequency points are used for smoothing, theimprovement from n′ to n″ can be quite helpful. For example, when theHann time-domain window is used, along with a similar raised-cosinefrequency-weighting shape for the smoothing, n″ ranges from 2.5n′ to9.1n′ as the number of smoothing points m ranges from 7 to 26. Thisspeeds up the measurement by a factor of 2.5 to 9.1 or corresponds to areduction in added uncorrelated bias of roughly 2.0 dB to 4.8 dB forcross-correlation measurements.

It is important to note that equation (34) holds only when thecomplex-valued function Ĝ_(ab) is smoothed. If the magnitude of Ĝ_(ab)is taken before smoothing, no cancellation of the uncorrelated noisewill take place, and (34) does not apply. Magnitude smoothing willresult in smoother-looking but less-accurate graphs than those fromvector smoothing.

In some embodiments, the spectral smoothing is used to redistribute thespectral data from a linear scale to a log scale. Thus, the smoothingfunction H={H_(i)} may be different at each frequency location thesmoothing function is applied. For example, the smoothing function maycover an extent in frequency that depends upon (e.g., is proportionalto) the frequency point at which the function is applied. Because ofthis, the coefficient set {d_(k)}, the coefficient a and the effectivenumber of averages n″ may also be different at each frequency for whichthe smoothed spectral data is computed.

Error Bounds

For the case of a single-channel measurement, we can accurately estimatethe uncertainty of V_(aa). Likewise, for the case of a dual-channelmeasurement when there is little or no uncorrelated noise in eachchannel, we can accurately estimate the error bounds of |V_(ab)|. Inboth cases, equation (20) above gives a robust estimate for the 95.45%confidence bounds and is repeated here, using n″ in place of n:

$\begin{matrix}{E_{95\%} = {{\pm 10}\; {\log \left( {\frac{2}{\sqrt{n^{''}} - 1} + 1} \right)}\mspace{14mu} {{dB}.}}} & (35)\end{matrix}$

(In the case where the data sets are overlapped but spectral smoothingis not performed, the above-described number n′ replaces n″ inexpression (35).) This approximation is accurate to within ±0.14 dB forn″≧1.7, and to within ±0.01 dB for n″≧18. Because of the cancellation inthe denominator, the formula is not useful for n″=1, and so it is bestto set n″=1.2 whenever n″<1.2. With that adjustment, the worst-caseerror for n″<1.7 is +2.0 dB at n″=1.2, where the expected error bound is±11.4 dB and the formula conservatively estimates ±13.4 dB.

On the other hand, creating a useful estimate of the error incross-spectral density measurements is not straightforward when there isa substantial amount of uncorrelated noise in each of the channels. Thestatistical distribution of the combination of all the terms of equation(12) becomes quite complicated, and the nature of the distributionchanges as the number of averages increases.

Given an average Ĝ_(ab) of n cross-spectral densities Ĝ_(ab), Bendat andPiersol offer the following error estimate for | Ĝ_(ab) |:

$\begin{matrix}{{{\sigma/\mu} = \frac{1}{{\hat{\gamma}}_{ab}\sqrt{n}}},} & (36)\end{matrix}$

where σ is the standard deviation of | Ĝ_(ab) |, where μ is the mean of| Ĝ_(ab) |, where {circumflex over (γ)}_(ab) is the square root of thecoherence function:

$\begin{matrix}{{\hat{\gamma}}_{ab}^{2} = {\frac{{{\overset{\_}{\hat{G}}}_{ab}}^{2}}{{\overset{\_}{\hat{G}}}_{aa}{\overset{\_}{\hat{G}}}_{bb}}.}} & (37)\end{matrix}$

(See Bendat, J. S., and Piersol, A. G., “Engineering Applications ofCorrelation and Spectral Analysis”, Wiley-Interscience, New York, 1980,pages 71-77.)

Bendat and Piersol do not suggest spectral smoothing. However, if were-interpret equation (36) in light of the above discussion of spectralsmoothing, we can replace the n occurring in equation (36) with n″:

$\begin{matrix}{{\sigma/\mu} = {\frac{1}{{\hat{\gamma}}_{ab}\sqrt{n^{''}}}.}} & \left( 36^{*} \right)\end{matrix}$

In this re-interpretation, the coherence function may be re-expressedas:

$\begin{matrix}{{{\hat{\gamma}}_{ab}^{2} = \frac{{V_{ab}}^{2}}{V_{aa}V_{bb}}},} & \left( 37^{*} \right)\end{matrix}$

where V_(aa) is the result of spectrally smoothing the mean Ĝ_(aa) ,where V_(bb) is the result of spectrally smoothing the mean Ĝ_(bb) .

The estimate (36) or (36*) suffers from several problems:

(a) The distributions are not normal (Gaussian), even when n″ is large.Hence σ by itself does not indicate very well what the 95.45% boundswould be.

(b) When n″=1, {circumflex over (γ)}_(ab) must also equal unity, and somust σ/μ. The true error, in fact, could be much higher if G_(aa) and/orG_(bb) are much larger than |G_(ab)|. This estimate of σ/μ remainserroneous for small values of n″ other than unity but improves withlarger values.

(c) Even for large values of n″, the error estimate σ remains roughlythe same if V_(aa) and/or V_(bb) are much larger than |V_(ab)|, nomatter how much larger they actually are, and so σ is not very helpful.

The following metric is offered, which for the most part corrects (a)and (b) and offers some improvement in (c).

Let

$\begin{matrix}{{S = {10\log \sqrt{\frac{n^{''}}{\frac{1}{{\hat{\gamma}}_{ab}^{2}} - 1 + ^{{({1 - n^{''}})}/20}}}}},} & (38)\end{matrix}$

where {circumflex over (γ)}_(ab) is defined in equation (37*). S is, ina sense, a signal-to-noise ratio metric which compares the correlatednoise to the uncorrelated noise and corrects for the effective number ofaverages taken. The exponential term prevents S from producing afalsely-optimistic estimate when n″ is low. Next, define

T=10^((0.81−0.1S+2000S) ⁻⁶ ₎.  (39)

T converts the signal-to-noise metric S into an approximation (in dB) ofcontribution of the uncorrelated noise to the 95.45% error bound. Thelarge S⁻⁶ term partially compensates for the tendency to underestimatethe error when little is known of the signal and the correlated noise ismuch larger than the uncorrelated noise. This mostly affects errorestimates greater than ±3 dB. Finally, define

$\begin{matrix}{{A = {10\; {\log \left( {\frac{2}{\sqrt{n^{''}} - 1} + 1} \right)}}},} & (40)\end{matrix}$

which is the same as equation (35) and accounts for the uncertainty inthe averaged correlated data |V_(yy)|, where V_(yy) represents theresult that would be obtained by spectrally smoothing the mean powerspectral density Ĝ_(yy) . As before, n″ should be bounded to ≧1.2.

Then the estimated 95.45% confidence error metric is:

E ₉₅%=±√{square root over (A ² +T ²)}dB.  (41)

This estimate will be referred to herein as EM (where EM stands forError Metric).

EM has been tested for accuracy. FIG. 5 illustrates the results of asimulation wherein 675000 sets of random correlation data were produced,with the number of averages ranging randomly from 1 to 50000 and theratio of the uncorrelated power to correlated power in each channelranging randomly from −20 dB to +30 dB. The X-axis is the EM 95%confidence metric in dB, and the Y-axis represents the 95.45% cumulativeprobability distribution of simulation results having the correspondingX-axis EM estimate. The ideal error metric would produce a straightdiagonal line, as shown by the dotted line in FIG. 5.

FIG. 5 shows that EM is very robust when the error is less than 3 dB.Above 3 dB, the estimate departs to the low side of the ideal curve,indicating somewhat conservative estimates. Were it not for the S⁻⁶term, it would branch considerably higher than the ideal curve,producing falsely lower error estimates.

It is emphasized that EM results above 3 dB are somewhat unreliable.When EM indicates less than 3 dB, the error distributions are fairlypredictable, but for EM above 3 dB, the distributions vary considerably.And while EM does a fairly good job of estimating the 95.45% error boundabove 3 dB, it does not indicate how large the remaining 4.55% of theerrors might be. Indeed, some of the simulated errors exceeded 20 dB inthis region.

The estimate EM is based on a 95% confidence probability. However, theprinciples described herein generalize to any confidence probability.

In some embodiments, the estimate EM may be determined without spectralsmoothing but with overlapping of sample data sets. In this case, whendetermining S and A, the coherence measure of expression (37) is usedinstead of the coherence measure of expression (37*), and the effectivenumber n′ is used instead of the effective number n″. In someembodiments, the estimate EM may be determined without spectralsmoothing and without overlapping of sample data sets. In this case,when determining S and A, the coherence measure of expression (37) isused instead of the coherence measure of expression (37*), and n is usedinstead of the effective number n″.

Method 600

In one set of embodiments, a method 600 for estimating a power spectraldensity of a target noise signal may include the operations shown inFIG. 6A. (Method 600 may also include any subset of the features,embodiments and operations described above.) The method 600 involvesvector smoothing, i.e., smoothing a complex-valued mean cross-spectraldensity in the frequency domain. Method 600 may be performed using acomputational device as described above.

Action 610 may include performing n iterations of a set of operations inorder to obtain n corresponding complex-valued cross spectral densitiesbetween a signal a(t) from a first channel and a signal b(t) from asecond channel, e.g., as described above. The signal a(t) is a sum of afirst interfering noise signal and the target noise signal; the signalb(t) is a sum of a second interfering noise signal and the target noisesignal. The integer n is greater than one. The set of operations mayinclude: (1) acquiring a set of samples of the signal a(t); (2)acquiring a set of samples of the second signal b(t), where the set ofsamples of the signal a(t) and the set of samples of the signal b(t) areacquired simultaneously (i.e., in parallel); and (3) computing acomplex-valued cross spectral density between the sample set of thesignal a(t) and the sample set of the signal b(t).

The integer n may reside in different ranges in different embodiments,or in different circumstances. In various circumstances, n may be,respectively, greater than 10, greater than 100, greater than 1000,greater than 10000, greater than 100,000, greater than 10⁶. In someembodiments, the integer n may be determined by user input.

Action 620 may include averaging the n complex-valued cross spectraldensities to obtain a complex-valued averaged cross spectral density. Insome embodiments, the average is an equally-weighted average. In otherembodiments, different ones of the cross spectral densities may beweighted differently.

Action 630 includes smoothing the complex-valued averaged cross spectraldensity in the frequency domain to obtain a complex-valued smoothedcross spectral density, e.g., as described above. The estimate of thepower spectral density G_(yy) of the target noise signal is based on thecomplex-valued smoothed cross spectral density. The estimate of G_(yy)may be determined by computing the magnitude of the smoothed crossspectral density.

Action 640 may include storing the estimate of the power spectraldensity of the target noise signal in a memory, e.g., a memory of thecomputational device, or a memory residing elsewhere, e.g., in a hostcomputer.

In some embodiments, the filter used to perform the smoothing varieswith frequency, e.g., as described above.

In some embodiments, the method 600 may also include displaying theestimate of the power spectral density of the target noise signal as agraph, e.g., as shown in FIG. 1A or 1B. In some embodiments, method 600may be applied to each product in an assembly line, to uniquelycharacterize the noise generated by each product. The power spectraldensity estimate might be stored in a memory medium and provided to acustomer along with the product.

The target noise signal y(t) may conform to any of various kinds ofnoise. In some embodiments, the target noise signal y(t) may be a phasenoise signal. In other embodiments, the target noise signal y(t) may bea voltage noise signal. In yet other embodiments, the target noisesignal y(t) may be a current noise signal. In yet other embodiments, thetarget noise signal y(t) may be an optical noise signal. In yet otherembodiments, the target noise signal y(t) may be a displacement noisesignal.

In some embodiments, the action of averaging the n complex-valuedcross-spectral densities comprises computing an updated average aftereach iteration of the set of operations, e.g., according to therelations:

SUM(m)=SUM(m−1)+Ĝ _(ab) ^((m))  (42A)

AVG(m)=(1/m)*SUM(m),  (42B)

where Ĝ_(ab) ^((m)) represents an m^(th) of the cross spectraldensities. In these embodiments, the smoothing may be performed on eachupdated average to obtain a corresponding smoothed cross spectraldensity. The magnitude of each smoothed cross spectral density may bedisplayed in succession so that a user can visualize how the uncertaintyin the power spectral density decreases as the number of averagesincreases.

In some embodiments, successive sets of the samples of the signal a(t)are overlapping in time, and successive sets of the samples of thesignal b(t) are overlapping in time, e.g., as described above.

In some embodiments, the set of operations includes: applying atime-domain window to the sample set of the signal a(t) to obtain afirst windowed sample set; and applying the time domain window to thesample set of the signal b(t) to obtain a second windowed sample set.See the above discussion regarding time-domain windowing. The action ofcomputing the complex-valued cross spectral density between the sampleset of the signal a(t) and the sample set of the signal b(t) comprisescomputing a complex-valued cross spectral density between the firstwindowed sample set and the second windowed sample set.

Method 650

In one set of embodiments, a method 650 for estimating a power spectraldensity of a target noise signal may include the actions shown in FIG.6B. (Method 650 may also include any subset of the features, embodimentsand operations described above.) Method 650 may be performed using acomputational device as described above.

Action 655 may include smoothing a complex-valued averaged crossspectral density to obtain a complex-valued smoothed cross spectraldensity, e.g., as variously described above. The complex-valued averagedcross spectral density is an average of n complex-valued cross spectraldensities, e.g., as variously described above. (The integer n is greaterthan one.) Each of the n complex-valued cross spectral densities iscomputed based on a corresponding one of n two-channel acquisitions,where each of the two-channel acquisitions includes an acquisition of aset of samples of a signal a(t) from a first channel and a simultaneousacquisition of a set of samples of a signal b(t) from a second channel,e.g., as variously described above. The signal a(t) is a sum of a firstinterfering noise signal and the target noise signal; the signal b(t) isa sum of a second interfering noise signal and the target noise signal.

Action 660 may include determining an estimate of the power spectraldensity of the target noise signal based on the complex-valued smoothedcross spectral density, e.g., as variously described above. The estimateof the power spectral density of the target noise signal is stored in amemory, e.g., a memory of the computational device.

In some embodiments, the filter used to perform the smoothing varieswith frequency, e.g., as described above.

In some embodiments, the method 650 may also include displaying theestimate of the power spectral density of the target noise signal as agraph.

The target noise signal y(t) may conform to any of various kinds ofnoise. In some embodiments, the target noise signal y(t) may be a phasenoise signal. In other embodiments, the target noise signal y(t) may bea voltage noise signal. In yet other embodiments, the target noisesignal y(t) may be a current noise signal. In yet other embodiments, thetarget noise signal y(t) may be an optical noise signal. In yet otherembodiments, the target noise signal y(t) may be a displacement noisesignal.

In some embodiments, the successive sets of samples of the signal a(t)are overlapping in time, and the successive sets of the samples of thesignal b(t) are overlapping in time, e.g., as variously described above.

In some embodiments, for each of the n two-channel acquisitions, thesample set of the signal a(t) is windowed with a time domain window toobtain a first windowed sample set, and the sample set of the signalb(t) is windowed with the time domain window to obtain a second windowedsample set, e.g., as variously described above. Each of the ncomplex-valued cross spectral densities is based on the first windowedsample set and the second windowed sample set of the correspondingtwo-channel acquisition, e.g., as variously described above.

In some embodiments, the method 650 also includes averaging the ncomplex-valued cross spectral densities to obtain the complex-valuedaveraged cross spectral density, e.g., as variously described above. Theaveraging may include computing an updated average after each of the ntwo-channel acquisitions, e.g., as variously described above.

Method 700

In one set of embodiments, a method 700 may include the operations shownin FIG. 7A. (Method 700 may also include any subset of the features,embodiments and operations described above.) Method 700 may be performedusing a computational device as described above.

Action 710 may include determining a statistical error bound for anestimate of a power spectral density of a target noise signal y(t),e.g., as variously described above. The power spectral density estimatemay be determined by averaging n complex-valued cross spectral densitiesto obtain a complex-valued averaged cross spectral density. Themagnitude of the complex-valued averaged cross spectral density may beused as the power spectral density estimate for the target noise signal.The n complex-valued cross spectral densities are computed based on nrespective two-channel data sets. (Each of the n complex-valued crossspectral densities may be computed in the traditional manner usingFourier transforms. However, it is noted that there are other means forcomputing cross spectral densities besides strict Fourier transforms,and the metrics and bounds described herein will apply to those othermeans as well.) Each two-channel data set includes a set of samples of asignal a(t) acquired from a first channel and a corresponding set ofsamples of a signal b(t) acquired from a second channel. Each set ofsamples of the signal a(t) and the corresponding set of samples of thesecond signal b(t) are acquired over the same interval of time. Thesignal a(t) is a sum of a first interfering noise signal and the targetnoise signal y(t). The signal b(t) is a sum of a second interferingnoise signal and the target noise signal y(t).

The action of determining the statistical error bound may includecomputing the statistical error bound based on an expression of the form√{square root over (A²+T²)}, where A depends on the number n, where Tdepends on the number n and a coherence function associated with thecomplex-valued averaged cross spectral density. For example, one maycompute A and T using expressions (37) and (38) through (40), but with nreplacing n″. Other cases will be discussed below.

It is noted that the statistical error bound is a function of frequencyat least because the coherence function is a function of frequency. Seeequations (37) and (37*).

Action 720 includes storing the statistical error bound in a memory,e.g., a memory of the computational device.

The term “statistical error bound” implies that the power spectraldensity of the target noise signal has a given probability P of beingwithin the interval of radius B centered on the power spectral densityestimate, where B is the statistical error bound. The probability P maybe different in different embodiments. In some embodiments, theprobability P may be determined by user input.

In some embodiments, the action of determining the statistical errorbound includes computing an effective number of independent averages n′corresponding to the power spectral density estimate based on dataincluding the number n and a relative amount of time overlap betweensuccessive ones of the sample sets of the signal a(t) and betweensuccessive ones of the sample sets of the signal b(t). In theseembodiments, A depends on the effective number n′, and T depends on theeffective number n′ and the coherence function. For example, one maycompute A and T using expression (37) and expressions (38) through (40),but with n′ replacing n″.

In some embodiments, the action of determining the power spectraldensity estimate includes spectrally smoothing the complex-valuedaveraged cross spectral density to obtain a spectrally smoothed crossspectral density, e.g., as described above. (The magnitude of thespectrally smoothed cross spectral density may used as the powerspectral density estimate.) In these embodiments, the action ofdetermining the statistical error bound may include computing aneffective number of independent averages n″ corresponding to the powerspectral density estimate based on data including: (1) the number n; (2)information specifying a time-domain window that is applied to thesample sets of the signal a(t) and to the sample sets of the signalb(t); and (3) information specifying the filter used to perform saidspectral smoothing. (The effective number n″ may be computed, e.g., asdescribed above in connection with equation (34).) Furthermore, Adepends on the effective number n″, and T depends on the effectivenumber n″ and on a coherence function associated with the spectrallysmoothed cross spectral density. For example, one may use expressions(37*) and (38) through (40) to compute A and T.

In some embodiments, the data used to compute the effective number n″may also include a relative amount of time overlap between successiveones of the sample sets of the signal a(t) and between successive onesof the sample sets of the signal b(t), e.g., as described above.

In some embodiments, the filter used to perform the spectral smoothingvaries with frequency, e.g., as described above in the section entitled“Spectral Smoothing”. In some embodiments, the spectral smoothingredistributes the complex-valued averaged cross spectral density from alinear frequency scale to a logarithmic frequency scale.

In some embodiments, the method 700 may also include displaying a graphof the estimate of the power spectral density of the target noise signaly(t). Furthermore, a graphical indication of the statistical error boundmay be displayed in association with the graph of the power spectraldensity estimate. For example, the graphical indication may be a set ofone or more error bars that are superimposed and/or centered on thegraph of the power spectral density estimate at one or more respectivefrequency locations. The size of the error bars correspond to themagnitude of the statistical error bound at the corresponding frequencylocations.

In one alternative embodiment, the graphical indication of thestatistical error bound includes a graph of an upper bound function anda graph of a lower bound function, where the upper bound function is thepower spectral density estimate plus the statistical error bound, andthe lower bound function is the power spectral density estimate minusthe statistical error bound.

The target noise signal y(t) may conform to any of various kinds ofnoise. In some embodiments, the target noise signal y(t) may be a phasenoise signal. In other embodiments, the target noise signal y(t) may bea voltage noise signal. In yet other embodiments, the target noisesignal y(t) may be a current noise signal. In yet other embodiments, thetarget noise signal y(t) may be an optical noise signal. In yet otherembodiments, the target noise signal y(t) may be a displacement noisesignal.

Method 800

In one set of embodiments, a method 800 may include the operations shownin FIG. 7B. (Method 800 may also include any subset of the features,embodiments and operations described above.) Method 800 may be performedusing a computational device as described above.

Action 810 may include determining a measure of uncertainty for anestimate of a power spectral density of a noise signal. The powerspectral density estimate may be determined by computing an average of npower spectral densities derived from n respective sets of samples ofthe noise signal. Each of the n power spectral densities is computedbased on a corresponding one of the sample sets. (Each of the n powerspectral densities may be computed in the traditional manner using aFourier transform. However, it is noted that there are other means forcomputing power spectral densities besides strict Fourier transforms,and the metrics and uncertainty measures described herein will apply tothose other means as well.)

The action of determining the uncertainty measure may include computingthe uncertainty measure based on the number n. For example, theuncertainty measure may be computed as described above in connectionwith expression (20). Other examples will be described below.

Action 820 may include storing the uncertainty measure in a memory.

In some embodiments, the method 800 may also include displaying a graphof the estimate of the power spectral density of the noise signal.Furthermore, a graphical indication of the uncertainty measure may bedisplayed in association with the displayed graph, e.g., as variouslydescribed above.

In some embodiments, the action of determining the uncertainty measureincludes computing an effective number of independent averages n′corresponding to the power spectral density estimate based on dataincluding: the number n and a relative amount of overlap betweensuccessive ones of the n sample sets, e.g., as described above. Theuncertainty measure may then be computed based on the effective numbern′.

In some embodiments, the action of determining the power spectraldensity estimate includes spectrally smoothing the average of the npower spectral densities to obtain a spectrally smoothed power spectraldensity (e.g., as described above in the section entitled “SpectralSmoothing”). In this case, the action of determining the uncertaintymeasure may include computing an effective number of independentaverages n″ corresponding to the power spectral density estimate basedon data including: (a) the number n; (b) information specifying atime-domain window that is applied to each of the n sample sets; and (c)information specifying the filter used to perform the spectralsmoothing. The uncertainty measure may then be computed based on theeffective number n″, e.g., as described above in connection withexpression (35).

In some embodiments, the data used to compute the effective number n″may also include a relative amount of overlap between successive ones ofthe n sample sets.

The target noise signal y(t) may conform to any of various kinds ofnoise. In some embodiments, the target noise signal y(t) may be a phasenoise signal. In other embodiments, the target noise signal y(t) may bea voltage noise signal. In yet other embodiments, the target noisesignal y(t) may be a current noise signal. In yet other embodiments, thetarget noise signal y(t) may be an optical noise signal. In yet otherembodiments, the target noise signal y(t) may be a displacement noisesignal.

Note that the metric given above in expression (41) specifies astatistical error range, i.e., both an upper bound and lower bound, sothat the actual noise density (of the target noise signal) is within theerror range with given probability (e.g., P=95%). In alternativeembodiments, instead of calculating an error range only an upper boundis calculated, where the actual noise density is lower than theestimated upper bound with probability equal to or greater than P (e.g.,P=95%). This upper bound is significantly lower than the upper boundprovided by the expression (41), especially when the data is poorlyresolved, which happens when the interfering noise is greater than thetarget noise (the noise to be measured). This alternative metric isuseful in production test applications, where knowing the exact noisedensity is not as important as knowing that the noise density is simplybelow some given amount, say, a test limit. It can significantlydecrease the amount of time required to test a product: one can stopaccumulating acquisitions when the upper bound falls below anestablished limit.

In one embodiment, the statistical upper bound is given by:

$\begin{matrix}{\left( G_{yy} \right)_{UB} = {\left( {{\frac{\sqrt{V_{aa}V_{bb}}}{n^{''}}\sqrt{1 + {\frac{\pi}{4}\left( {n^{''} - 1} \right)}}} + {V_{ab}}} \right)\left( {\frac{1}{\sqrt{n^{''}} - 1} + 1} \right)}} & (43)\end{matrix}$

where V_(aa), V_(bb), V_(ab) and n″ are as defined above, where

G _(yy)≦(G _(yy))_(UB)  (44)

with probability greater than or equal to 95%. Thus, the statisticalupper bound is a product of two terms, one depending on n″ and the otherdepending on n″, V_(aa), V_(bb) and V_(ab).

If spectral smoothing is not performed, then V_(aa), V_(bb), V_(ab) andn″ are replaced by Ĝ_(aa) , Ĝ_(bb) , Ĝ_(ab) and n′ (or n if successivesample sets are not overlapped).

The statistical upper bound (43) assumes that the power spectral densityestimate for the target noise signal y(t) is based on two-channelacquisitions. However, as described above, a power spectral densityestimate for the total noise signal (i.e., the sum of the target noisesignal plus any interfering noise signal) may be computed based onsingle-channel acquisitions. A statistical upper bound for this case maybe computed based on the expression:

$\begin{matrix}{{\left( G_{aa} \right)_{UB} = {V_{aa}\left( {\frac{1}{\sqrt{n^{''}} - 1} + 1} \right)}},} & (45)\end{matrix}$

where V_(aa) and n″ are as defined above, where

G _(aa)≦(G _(aa))_(UB)  (46)

with probability greater than or equal to 95%. If spectral smoothing isnot performed, then V_(aa) and n″ are replaced by Ĝ_(aa) and n′ (or n ifsuccessive sample sets are not overlapped).

Method 850

In one set of embodiments, a method 850 may include the operations shownin FIG. 8. (Method 850 may also include any subset of the features,embodiments and operations described above.) Method 850 may be performedusing a computational device as described above.

Action 855 may include determining a statistical upper bound (e.g., asdescribed above) for an estimate of a power spectral density of a targetnoise signal y(t). The estimate of the power spectral density isdetermined by averaging n complex-valued cross spectral densities toobtain a complex-valued averaged cross spectral density. The ncomplex-valued cross spectral densities are computed based on nrespective two-channel data sets. (Each of the n complex-valued crossspectral densities may be computed in the traditional manner usingFourier transforms. However, it is noted that there are other means forcomputing cross spectral densities besides strict Fourier transforms,and the metrics and bounds described herein will apply to those othermeans as well.) Each two-channel data set includes a set of samples of asignal a(t) acquired from a first channel and a corresponding set ofsamples of a signal b(t) acquired from a second channel. Each set ofsamples of the signal a(t) and the corresponding set of samples of thesecond signal b(t) are acquired over the same interval of time. Thesignal a(t) is a sum of a first interfering noise signal and the targetnoise signal y(t). The signal b(t) is a sum of a second interferingnoise signal and the target noise signal y(t).

The action of determining the statistical upper bound may includecomputing the statistical upper bound based on the number n, thecomplex-valued averaged cross spectral density, a first spectral densityand a second spectral density, where the first spectral density is anaveraged power spectral density for the signal a(t), where the secondspectral density is an averaged power spectral density for the signalb(t). For example, the statistical upper bound may be computed asindicated by expression (43), but with n replacing n″ and with Ĝ_(aa) ,Ĝ_(bb) , Ĝ_(ab) replacing V_(aa), V_(bb) and V_(ab). Other examples arediscussed below.

Action 870 may include storing the statistical upper bound in a memory.

In some embodiments, the method 850 may also include: displaying a graphof the estimate of the power spectral density of the target noise signaly(t); and displaying the statistical upper bound in association with thegraph, e.g., as variously described above.

In some embodiments, the action of determining the statistical upperbound includes computing an effective number of independent averages n′corresponding to the power spectral density estimate based on dataincluding: the number n and a relative amount of time overlap betweensuccessive ones of the sample sets of the signal a(t) and betweensuccessive ones of the sample sets of the signal b(t). The statisticalupper bound may then be computed based on the effective number n′, thecomplex-valued averaged cross spectral density, the first spectraldensity and the second spectral density (e.g., as indicated by theexpression (43), but with n′ replacing n″ and with Ĝ_(aa) , Ĝ_(bb) ,Ĝ_(ab) replacing V_(aa), V_(bb) and V_(ab)).

In some embodiments, the action of determining the power spectraldensity estimate includes spectrally smoothing the complex-valuedaveraged cross spectral density to obtain a spectrally smoothed crossspectral density. (The magnitude of the spectrally smoothed crossspectral density may be used as the power spectral density estimate.) Inthis case, the action of determining the statistical upper bound mayinclude computing an effective number of independent averages n″corresponding to the power spectral density estimate based on dataincluding: (a) the number n; (b) information specifying a time-domainwindow that is applied to the sample sets of the signal a(t) and to thesample sets of the signal b(t); and (c) information specifying thefilter used to perform the spectral smoothing. The statistical upperbound may then be computed based on the effective number n″, thespectrally smoothed cross spectral density, and spectrally smoothedversions of the first and second spectra, e.g., as indicated byexpression (43).

In some embodiments, the filter used to perform the spectral smoothingvaries with frequency, e.g., as described above.

In some embodiments, the spectral smoothing redistributes thecomplex-valued averaged cross spectral density from a linear frequencyscale to a logarithmic frequency scale.

In some embodiments, the target noise signal y(t) is a phase noisesignal or a voltage noise signal or a current noise signal or an opticalnoise signal or a displacement noise signal.

In some embodiment, the statistical upper bound is computed based on theexpression:

${\left( G_{yy} \right)_{UB} = {\left( {{\frac{\sqrt{V_{aa}V_{bb}}}{n^{''}}\sqrt{1 + {\frac{\pi}{4}\left( {n^{''} - 1} \right)}}} + {V_{ab}}} \right)\left( {\frac{1}{\sqrt{n^{''}} - 1} + 1} \right)}},$

where n″ is the effective number of averages.

Computer System

FIG. 9 illustrates one embodiment of a computer system 900 that may beused to perform any of the method embodiments described herein, or, anycombination of the method embodiments described herein, or any subset ofany of the method embodiments described herein, or, any combination ofsuch subsets.

Computer system 900 may include a processing unit 910, a system memory912, a set 915 of one or more storage devices, a communication bus 920,a set 925 of input devices, and a display system 930.

System memory 912 may include a set of semiconductor devices such as RAMdevices (and perhaps also a set of ROM devices).

Storage devices 915 may include any of various storage devices such asone or more memory media and/or memory access devices. For example,storage devices 915 may include devices such as a CD/DVD-ROM drive, ahard disk, a magnetic disk drive, magnetic tape drives, etc.

Processing unit 910 is configured to read and execute programinstructions, e.g., program instructions stored in system memory 912and/or on one or more of the storage devices 915. Processing unit 910may couple to system memory 912 through communication bus 920 (orthrough a system of interconnected busses). The program instructionsconfigure the computer system 900 to implement a method, e.g., any ofthe method embodiments described herein, or, any combination of themethod embodiments described herein, or, any subset of any of the methodembodiments described herein, or any combination of such subsets.

Processing unit 910 may include one or more processors (e.g.,microprocessors).

One or more users may supply input to the computer system 900 throughthe input devices 925. Input devices 925 may include devices such as akeyboard, a mouse, a touch-sensitive pad, a touch-sensitive screen, adrawing pad, a track ball, a light pen, a data glove, eye orientationand/or head orientation sensors, a microphone (or set of microphones),or any combination thereof.

The display system 930 may include any of a wide variety of displaydevices representing any of a wide variety of display technologies. Forexample, the display system may be a computer monitor, a head-mounteddisplay, a projector system, a volumetric display, or a combinationthereof. In some embodiments, the display system may include a pluralityof display devices. In one embodiment, the display system may include aprinter and/or a plotter.

In some embodiments, the computer system 900 may include other devices,e.g., devices such as one or more graphics accelerators, one or morespeakers, a sound card, a video camera and a video card.

In some embodiments, computer system 900 may include one or morecommunication devices 935, e.g., a network interface card forinterfacing with a computer network.

In some embodiments, the communication devices may include areconfigurable I/O (RIO) board that includes one or more programmablehardware elements (PHEs), one or more A/D converters and perhapsadditional circuitry. The RIO board is programmable to achieve auser-desired configuration of input and/or output processing, e.g., viaa program written using LabVIEW FPGA. In some embodiments, thereconfigurable I/O board is one of the RIO boards provided by NationalInstruments Corporation.

The computer system may be configured with a software infrastructureincluding an operating system, one or more compilers for one or morecorresponding programming languages, and perhaps also one or moregraphics APIs (such as OpenGL®, Direct3D, Java 3D™). Any or all of thecompilers may be configured to perform expression rearrangementaccording to any or all of the method embodiments described herein. Insome embodiments, the software infrastructure may include LabVIEW and/orLabVIEW FPGA, which are software products of National InstrumentsCorporation.

In some embodiments, the computer system 900 may be configured forcoupling to a data acquisition system 940. The data acquisition system940 is configured to receive analog inputs signals, to digitize theanalog input signals, and to make those digitized signals available tothe computer system 900. The data acquisition system 940 may operateunder the control of the software executing on processor 910. In someembodiments, the data acquisition system 940 includes two A/D convertersthat are configured to capture samples of two input signals in parallel,e.g., driven by the same sample conversion clock. The captured samplesmay be stored into a memory of the data acquisition system 940 and madeavailable for access by host software executing on processing unit 910.

FIG. 10 illustrates one possible embodiment 1000 for computer system900.

What is claimed is:
 1. A method comprising: acquiring n two-channel datasets and storing the n two-channel data sets in a memory, wherein foreach two-channel data set said acquiring comprises: acquiring a set ofsamples of a signal a(t) from a first channel, wherein the signal a(t)is a sum of a first interfering noise signal and a target noise signaly(t); acquiring a set of samples of a signal b(t) from a second channel,wherein the signal b(t) is a sum of a second interfering noise signaland the target noise signal y(t), wherein the sample set of the signala(t) and the sample set of the signal b(t) are acquired over the sameinterval of time; utilizing a computational device to implement:determining a statistical error bound for an estimate of a powerspectral density of a target noise signal y(t), wherein the estimate isdetermined by averaging n complex-valued cross spectral densities toobtain a complex-valued averaged cross spectral density, wherein the ncomplex-valued cross spectral densities are computed based respectivelyon the n two-channel data sets, wherein said determining the statisticalerror bound includes computing the statistical error bound based on anexpression of the form √{square root over (A²+T²)}, wherein A depends onthe number n, wherein T depends on the number n and a coherence functionassociated with the complex-valued averaged cross spectral density;storing the statistical error bound in a memory.
 2. The method of claim1, further comprising: displaying a graph of the estimate of the powerspectral density of the target noise signal y(t); and displaying agraphical indication of the statistical error bound in association withthe graph.
 3. The method of claim 1, wherein said determining thestatistical error bound includes computing an effective number ofindependent averages corresponding to said power spectral densityestimate based on data including the number n and a relative amount oftime overlap between successive ones of the sample sets of the signala(t) and between successive ones of the sample sets of the signal b(t),wherein A depends on the effective number, wherein T depends on theeffective number and the coherence function.
 4. The method of claim 1,wherein said determining the power spectral density estimate includesspectrally smoothing the complex-valued averaged cross spectral densityto obtain a spectrally smoothed cross spectral density, wherein saiddetermining the statistical error bound includes computing an effectivenumber of independent averages corresponding to said power spectraldensity estimate based on data including: the number n; informationspecifying a time-domain window that is applied to the sample sets ofthe signal a(t) and to the sample sets of the signal b(t); andinformation specifying the filter used to perform said spectralsmoothing, wherein A depends on the effective number, wherein T dependson the effective number and on a coherence function associated with thespectrally smoothed cross spectral density.
 5. The method of claim 4,wherein the filter used to perform the spectral smoothing varies withfrequency.
 6. The method of claim 5, wherein the spectral smoothingredistributes the complex-valued averaged cross spectral density from alinear frequency scale to a logarithmic frequency scale.
 7. A systemcomprising: a memory configured to store program instructions; and aprocessor configured to access the program instructions from the memoryand execute the program instructions, wherein the program instructions,when executed by the processor, cause the processor to implement:determining a statistical error bound for an estimate of a powerspectral density of a target noise signal y(t), wherein the estimate isdetermined by averaging n complex-valued cross spectral densities toobtain a complex-valued averaged cross spectral density, wherein the ncomplex-valued cross spectral densities are computed based on nrespective two-channel data sets, wherein each two-channel data setincludes a set of samples of a signal a(t) acquired from a first channeland a corresponding set of samples of a signal b(t) acquired from asecond channel, wherein each set of samples of the signal a(t) and thecorresponding set of samples of the second signal b(t) are acquired overthe same interval of time, wherein the signal a(t) is a sum of a firstinterfering noise signal and the target noise signal y(t), wherein thesignal b(t) is a sum of a second interfering noise signal and the targetnoise signal y(t), wherein said determining the statistical error boundincludes computing the statistical error bound based on an expression ofthe form √{square root over (A²+T²)}, wherein A depends on the number n,wherein T depends on the number n and a coherence function associatedwith the complex-valued averaged cross spectral density; storing thestatistical error bound in a memory medium.
 8. The system of claim 7,wherein the program instructions, when executed by the processor,further cause the processor to implement: displaying a graph of theestimate of the power spectral density of the target noise signal y(t);and displaying a graphical indication of the statistical error bound inassociation with the graph.
 9. The system of claim 7, wherein saiddetermining the statistical error bound includes computing an effectivenumber of independent averages corresponding to said power spectraldensity estimate based on data including the number n and a relativeamount of time overlap between successive ones of the sample sets of thesignal a(t) and between successive ones of the sample sets of the signalb(t), wherein A depends on the effective number, wherein T depends onthe effective number and the coherence function.
 10. The system of claim7, wherein said determining the power spectral density estimate includesspectrally smoothing the complex-valued averaged cross spectral densityto obtain a spectrally smoothed cross spectral density, wherein saiddetermining the statistical error bound includes computing an effectivenumber of independent averages corresponding to said power spectraldensity estimate based on data including: the number n; informationspecifying a time-domain window that is applied to the sample sets ofthe signal a(t) and to the sample sets of the signal b(t); andinformation specifying the filter used to perform said spectralsmoothing, wherein A depends on the effective number, wherein T dependson the effective number and on a coherence function associated with thespectrally smoothed cross spectral density.
 11. The system of claim 10,wherein the filter used to perform the spectrally smoothing varies withfrequency.
 12. The system of claim 11, wherein the smoothingredistributes the complex-valued averaged cross spectral density from alinear frequency scale to a logarithmic frequency scale.
 13. A methodcomprising: utilizing a computational device to implement: determining ameasure of uncertainty for an estimate of a power spectral density of anoise signal, wherein the estimate is determined by computing an averageof n power spectral densities derived from n respective sets of samplesof the noise signal, wherein each of the n power spectral densities iscomputed based on a corresponding one of the sample sets, wherein saiddetermining the uncertainty measure includes computing the uncertaintymeasure based on the number n; and storing the uncertainty measure in amemory.
 14. The method of claim 13, wherein said determining theuncertainty measure includes computing an effective number ofindependent averages corresponding to said power spectral densityestimate based on data including: the number n and a relative amount ofoverlap between successive ones of the n sample sets, wherein theuncertainty measure is computed based on the effective number ofindependent averages.
 15. The method of claim 13, wherein saiddetermining the power spectral density estimate includes spectrallysmoothing the average of the n power spectral densities to obtain aspectrally smoothed power spectral density, wherein said determining theuncertainty measure includes computing an effective number ofindependent averages corresponding to said power spectral densityestimate based on data including: the number n; information specifying atime-domain window that is applied to each of the n sample sets; andinformation specifying the filter used to perform said spectralsmoothing; wherein the uncertainty measure is computed based on theeffective number of independent averages.
 16. The method of claim 13,further comprising: utilizing the computational device to furtherimplement: displaying a graph of the estimate of the power spectraldensity of the noise signal; and displaying a graphical indication ofthe uncertainty measure in association with the displayed graph.
 17. Asystem comprising: a memory configured to store program instructions;and a processor configured to access the program instructions from thememory and execute the program instructions, wherein the programinstructions, when executed by the processor, cause the processor toimplement: determining a measure of uncertainty for an estimate of apower spectral density of a noise signal, wherein the estimate isdetermined by computing an average of n power spectral densities derivedfrom n respective sets of samples of the noise signal, wherein each ofthe n power spectral densities is computed based on a corresponding oneof the sample sets, wherein said determining the uncertainty measureincludes computing the uncertainty measure based on the number n;storing the uncertainty measure in a memory medium.
 18. The system ofclaim 17, wherein the program instructions, when executed by theprocessor, further cause the processor to implement: displaying a graphof the estimate of the power spectral density of the noise signal; anddisplaying a graphical indication of the uncertainty measure inassociation with the displayed graph.
 19. The system of claim 17,wherein said determining the uncertainty measure includes computing aneffective number of independent averages corresponding to said powerspectral density estimate based on data including: the number n and arelative amount of overlap between successive ones of the n sample sets,wherein the uncertainty measure is computed based on the effectivenumber of independent averages.
 20. The system of claim 17, wherein saiddetermining the power spectral density estimate includes spectrallysmoothing the average of the n power spectral densities to obtain aspectrally smoothed power spectral density, wherein said determining theuncertainty measure also includes computing an effective number ofindependent averages corresponding to said power spectral densityestimate based on data including: the number n; information specifying atime-domain window that is applied to each of the n sample sets; andinformation specifying the filter used to perform said spectralsmoothing; wherein the uncertainty measure is computed based on theeffective number of independent averages.